A Converse Comparison Theorem for BSDEs and Related Properties of g-Expectation

Briand, Philippe; Coquet, François; Hu, Ying; Mémin, Jean; Péng, Shigē

doi:10.1214/ecp.v5-1025

Cited by 144 publications

(187 citation statements)

References 9 publications

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“…It is a nonlinear mapping, but it preserves almost all other properties of the classical linear expectations. For more detailed views on this topic, we refer to [11], [4], [12], or [1] where some special cases are studied in depth, including the y-independent case, which will turn out to be the natural setting behind the present work. For applications of g-expectations to utility theory in economics, we refer to [3].…”

Section: Existsmentioning

confidence: 99%

Filtration-consistent nonlinear expectations and related g-expectations

Coquet

Mémin

et al. 2002

Probab Theory Relat Fields

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From a general definition of nonlinear expectations, viewed as operators preserving monotonicity and constants, we derive, under rather general assumptions, the notions of conditional nonlinear expectation and nonlinear martingale. We prove that any such nonlinear martingale can be represented as the solution of a backward stochastic equation, and in particular admits continuous paths. In other words, it is a g-martingale.

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Section: Existsmentioning

confidence: 99%

Filtration-consistent nonlinear expectations and related g-expectations

Coquet

Mémin

et al. 2002

Probab Theory Relat Fields

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221

267

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“…These were introduced by Peng, [14]. Basic papers explaining the relation with time consistent risk measures are [6], [8].…”

Section: Introduction and Notationmentioning

confidence: 99%

The Structure of m–Stable Sets and in Particular of the Set of Risk Neutral Measures

Delbaen

2006

Lecture Notes in Mathematics

156

221

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Abstract. The study of dynamic coherent risk measures and risk adjusted values as introduced by Artzner, Delbaen, Eber, Heath and Ku, leads to a property called fork convexity, rectangularity or m-stability. We give necessary and sufficient conditions for a closed convex set of measures to be fork convex. Since the set of martingale measures for price processes is m-stable, this leads to a characterisation of closed convex sets that can be obtained as the set of risk neutral measures in an arbitrage free model of security prices. We also relate the property of m-stability with the validity of Bellman's principle. It turns out that the stability property investigated in this paper is equivalent to properties known as time-consistency and rectangularity as used in multiprior Bayesian decision theory. Finally we characterise the set of risk neutral measures for continuous price processes.

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“…This problem was initiated in Briand et al [2] in which a counterexample was given to show that the above generalized Jensen's inequality fails for a very simple convex function h, and they gave a sufficient condition for a special situation. Chen et al [6] obtained a very interesting result: if g does not depend on y, the above generalized Jensen's inequality holds true for each convex function h if and only if g is a superhomogeneous function, i.e., g(t, λz) ≥ λg(t, z), d P × dt − a.s. for λ ∈ R and z ∈ R d .…”

Section: Holds For Each Random Variable X Such That Both E[x ] and E[mentioning

confidence: 99%

Jensen’s inequality for g-convex function under g-expectation

Jia

Péng

2009

Probab. Theory Relat. Fields

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A real valued function h defined on R is called g-convex if it satisfies the "generalized Jensen's inequality" for a given g-holds for all random variables X such that both sides of the inequality are meaningful. In this paper we will give a necessary and sufficient condition for a C 2 -function being g-convex, and study some more general situations. We also study g-concave and g-affine functions, and a relation between g-convexity and backward stochastic viability property.

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A Converse Comparison Theorem for BSDEs and Related Properties of g-Expectation

Cited by 144 publications

References 9 publications

Filtration-consistent nonlinear expectations and related g-expectations

Filtration-consistent nonlinear expectations and related g-expectations

The Structure of m–Stable Sets and in Particular of the Set of Risk Neutral Measures

Jensen’s inequality for g-convex function under g-expectation

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